Uncertainty Restrictions on Orbital Angular Momentum

The general uncertainty relation of quantum mechanics forbids the simultaneous and completely accurate measurements of two observables whose operators do not commute. As will be shown here, the one-dimensional orbital angular momentum operators do not mutually commute. It follows then that, for example, one cannot suggest a system in which there is non-zero orbital angular momentum in only one dimension of orientation, as the knowledge of the orbital angular momentum in said dimension would imply that it is impossible to know that there is zero orbital angular momentum in the other two dimensions. Certain restrictions can thus be derived from the general uncertainty relation for orbital angular momentum.